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Preprint submitted on 3 Apr 2013
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Thomas Batard, Michel Berthier
To cite this version:
Thomas Batard, Michel Berthier. Spinor Fourier Transform for Image Processing. 2013. �hal- 00807265�
Spinor Fourier Transform for Image Processing
Thomas Batard, Michel Berthier
Abstract—We propose in this paper to introduce a new spinor Fourier transform for both grey-level and color image processing.
Our approach relies on the three following considerations:
mathematically speaking, defining a Fourier transform requires to deal with group actions; vectors of the acquisition space can be considered as generalized numbers when embedded in a Clifford algebra; the tangent space of the image surface appears to be a natural parameter of the transform we define by means of so-called spin characters. The resulting spinor Fourier transform may be used to perform frequency filtering that takes into account the Riemannian geometry of the image. We give examples of low-pass filtering interpreted as diffusion process.
When applied to color images, the entire color information is involved in a really non marginal process.
Index Terms—Fourier transform, Scale-space, Riemannian geometry, Grey-level image, Color image.
I. INTRODUCTION
FROM the mathematical viewpoint, defining a Fourier transform requires the use of group representations (see Appendix B or [28]). Let us illustrate this fact on the well known shift theorem. Iff :R−→Ris a function that admits a Fourier transform andfα:R−→Ris defined by
fα(x) =f(x+α) (1) then the Fourier transforms F(f)andF(fα)of f andfα are linked by
F(fα)(u) =e2iπαuF(f) (2) The group involved here is the additive group (R,+) of translations acting on Rby
(α, x)7−→x+α:=τα(x) (3) The correspondence
χu:τα7−→e2iπαu=χu(α) (4) is a composition law preserving map from the group (R,+) to the group S1 of unit complex numbers acting on C by multiplication. It is a one-dimensional representation, i.e. a character, of the group (R,+). It appears that the Fourier transform is defined on the set of characters, also called the Pontryagin dual, by
F(f)(u) = Z
R
f(x)χu(−x)dx (5) identifying χu with u. The rest of this paper consists in showing how to generalize the above formula (5) so as to define a spinor Fourier transform well adapted to grey-level and color image processing.
Let us first formulate some remarks.
Thomas Batard, Lab. XLIM-SIC, University of Poitiers, France.
Michel Berthier, Lab. MIA, University of La Rochelle, France.
1) Marginal, i.e. componentwise, processing applied on color images is well known to produce false colors (see for instance [23] for examples of false colors). But it is not so clear when dealing with such images, how to define a Fourier transform which does not reduce to three Fourier transforms computed marginally and that takes into account color data.
2) The mathematical framework previously described also applies for a large range of groups. Considering the abelian finite group Z/NZ, resp. the abelian rotation group SO(2,R), leads to the definition of the discrete Fourier transform, resp. the definition of the Fourier series. The non abelian case which requires more math- ematical developments has been considered for instance in [27] to define generalized Fourier descriptors using the motion group of the plane and in [10] for the construction of so-called shearlets.
3) Very few works are devoted to a mixed approach of signal processing involving both harmonic and Rieman- nian methods. Although geometric decompositions such as curvelets take into account structure data (like edges), they are not directly defined by means of the Riemannian properties of the image surface.
4) As we are mainly concerned by grey-level and color image processing, it is important to note that the spinor Fourier transform we want to define must be computed with usual complex fast Fourier transforms.
Let us now describe the main underlying key ideas of this work. The first one is to extend the usual approach mentioned above to n-dimensional images by replacing the involved complex characters by characters with values in a group acting on the acquisition n-dimensional vector space. One way to proceed is to treat vectors as generalized numbers. This can be done by embedding the acquisition space in an algebra where are defined the four usual operations(+,−,×, /)(the inverse is defined only for invertible elements). As examples, it is well known that vectors ofR2, resp. vectors ofR4, can be identified with complex, resp. hypercomplex (quaternion) numbers (in both cases all non zero elements are invertible). We propose here to deal with Clifford algebras of which the complex and quaternion algebras appear to be particular cases. These algebras have already been used in many works related to signal processing (see [17], [14], [3], [2] or [13] for instance).
We are led to consider spin characters, that is composition law preserving maps from the group(R2,+) with values in spin groups. These latters act on vector spaces through the standard representations
ρn:Spin(n)−→SO(n,R) (6) that describe the groupsSpin(n)as double-sheeted coverings of the orthogonal groups SO(n,R)(see [25]). By averaging
this action we define a Clifford Fourier transform which appears to generalize in a very natural way the formula (5).
The second idea is to involve explicitly the geometry of the image surface that is embedded for instance in R3, resp.
R5 when considering grey-level, resp. color images. The al- ready mentioned spin characters are parametrized by so-called bivectors (see [20]). Very roughly speaking these bivectors, which are elements of the Clifford algebra, encode planes of R3 or R5. By using bivectors corresponding to the tangent planes of the image surface, one can parametrize the Clifford Fourier transform so as to take into account the Riemannian geometry of the image surface. To set up this idea one has to consider images as sections of associated vector bundles built from the standard representations of spin groups (see [18]).
These bundles are called spinor bundles and the corresponding sections spinor fields.
Finally, we introduce a Riemannian harmonic decompo- sition for images. The usual 2D discrete Fourier transform provides a basis of decomposition for say grey-level images given by
(m, n)7−→e2iπ(um/M+vn/N) (7) with u ∈ Z/MZ and v ∈ Z/NZ, in such a way that any imagef of sizeM ×N can be written as
f(m, n) =X
fb(u, v) e2πi(um/M+vn/N) (8) the sum running over m ∈ Z/MZ and n ∈ Z/NZ. The main drawbacks of this basis are that this one neither involves local geometric data nor extends in a non marginal way to multi-channel images. Using the spinor Fourier transform and the Riemannian settings described above allows one to obtain a decomposition basis for the image surface which takes into account both local geometric and color information. To illustrate this approach, we perform frequency filterings and show applications on well known images.
Let us mention that in this paper we consider the only standard representations (6) of the spin groups. Other represen- tations exist beside these which are called spin representations and that come from the complex representations of Clifford algebras. These representations do not descend to orthogonal groups (see [21]). The study of the corresponding Riemannian harmonic decomposition will appear elsewhere.
II. CLIFFORDFOURIERTRANSFORMS
THERE have been many attempts to generalize the usual Fourier transform in the framework of quaternion or Clifford algebras. We mention here briefly some of the already existing definitions and describe the construction involving the spin characters. The reader will find in the appendices the mathematical definitions and results used here.
A. Quaternionic and Clifford Fourier transforms
The quaternionic transform introduced in [26] (see also [15]) reads
Fµf(U) = Z
R2
f(X) exp(−µhX, Ui)dX (9)
X = (x1, x2), U = (u1, u2), and is defined for a function (color image)f :R2−→H0, whereH0is the set of imaginary quaternions. The unit imaginary quaternionµis chosen to be µ= (i+j+k)/√
3(representing the grey axis) and satisfies µ2=−1. The corresponding quaternionic Fourier coefficients are decomposed with respect to a symplectic split associated toµ, each one of the factors being expressed in the polar form:
Fµf =Akexp[µθk] +A⊥exp[µθ⊥]ν (10) withν an imaginary unit quaternion orthogonal toµ. The au- thors propose a spectral interpretation from this decomposition (see [26] for details).
In [7] (see also [8]) the approach is quite different since it concerns mainly the analysis of symmetries of a signalf from R2 toR given for example by a grey-level image. For such a signal, the quaternionic Fourier transform introduced in [7]
reads:
Fijf(U) = Z
R2
exp(−2πiu1x1)f(X) exp(−2πju2x2)dX (11) Note that i andj can be replaced by arbitrary pure imag- inary quaternions. The choice of this formula is justified by the following equality:
Fijf(U) =Fccf(U)−iFscf(U)−jFcsf(U) +kFssf(U) (12) where
Fccf(U) = Z
R2
f(X) cos(2πu1x1) cos(2πu2x2)dX (13) and similar expressions involving sines and cosines forFscf, Fcsf andFssf.
It is important to note at this stage that the kernels used in (9) and (11) correspond to rotations in the spaceR4under the identification of the groupSpin(4) with the group H1×H1 whereH1 is the group of unit quaternions (see Appendix A or [22]).
In [16], the Clifford Fourier transform is defined by Fe1e2f(U) =
Z
R
exp(−2πe1e2hU, Xi)f(X)dX (14) for a function f(X) =f(x)e2 fromRtoRand by
Fe1e2e3f(U) = Z
R2
exp(−2πe1e2e3hU, Xi)f(X)dX (15) for a functionf(X) =f(x1, x2)e3fromR2toR. The element e1e2, resp.e1e2e3, is the so-called pseudoscalar of the Clifford algebra R2,0, resp. R3,0. These transforms appear naturally when dealing with the analytic and monogenic signals.
The definition of [14] also uses the kernel of (15). If f is a function fromR3 to the Clifford algebraR3,0, then
Fe1e2e3f(U) = Z
R3
f(X) exp(−2πe1e2e3hU, Xi)dX (16) Note that if we set
f =f0+f1e1+f2e2+f3e3+
f23i3e1+f31i3e2+f12i3e3+f123i3 (17)
with i3 = e1e2e3, this transform can be written as a sum of four complex Fourier transforms by identifying i3 with the imaginary complex i. In particular, for a function f with values in the vector part of the Clifford algebra, this reduces to marginal, i.e. componentwise, processing. This Clifford Fourier transform is used to analyze frequencies of vector fields and the behavior of vector valued filters.
The definition proposed in [3] relies on Clifford analysis and involves the so-called angular Dirac operator Γ. The general formula is
F±f(U) = 1
√2π nZ
Rn
exp(∓iπ
2ΓU)×exp(−ihU, Xi)f(X)dX (18) For the special case of a functionf fromR2toC2=R0,2⊗C, the transform reads
F±f(U) = 1 2π
Z
R2
exp(±U∧X)f(X)dX (19) Let us remark that exp(±U ∧X) is the exponential of a bivector, i.e. a spinor. This construction allows to introduce two dimensional Clifford Gabor filters.
Let us mention that the above transform (19) is a very special case of a more general transform defined by means of Clifford analysis. We refer the reader to [4], [11] and [12]
for details. Finally, alternative definitions can be found in [5]
and [6] based on Clifford generalizations of the squared root of -1.
B. Clifford Fourier transform with spin characters
The transform defined in [1] is based on a spin generaliza- tion of the usual notion of group characters (see Sec. III.C).
If f is a function from R2 with values in the vector part of the Clifford algebra R4,0 then
CFf(u1, u2, u3, u4, D) = Z
R2
f(x1, x2)⊥ϕ(u1,u2,u3,u4,D)(−x1,−x2)dx1dx2 (20) with
(x1, x2)7−→ϕ(u1,u2,u3,u4,D)(x1, x2) (21) the spin character that sends (x1, x2)to the product
exp 1
2[x1(u1+u3) +x2(u2+u4)]D
×exp 1
2[x1(u1−u3) +x2(u2−u4)]I4D
(22) whereDis a unit bivector ofR4,0,I4is the pseudo scalar and
⊥is the action of Spin(4) onR4 (see Appendix A). For the special case of a color image
f : (x1, x2)7−→f1(x1, x2)e1+f2(x1, x2)e2+f3(x1, x2)e3 (23) the Clifford Fourier transform of the functionf in the direction D is given by
CFDf(u1, u2) = Z
R2
f(x1, x2)⊥ϕ(u1,u2,0,0,D)(−x1,−x2)dx1dx2 (24)
Note that formulas (20) and (24) appear as natural gener- alizations of the usual 2D Fourier transform for anR-valued function written as
Ff(u1, u2) = Z
R2
f(x1, x2)⊥ϕ(u1,u2,e1e2)(−x1,−x2)dx1dx2 (25) where
f(x1, x2) =f1(x1, x2)e1+f2(x1, x2)e2 (26) and
ϕ(u1,u2,e1e2)(x1, x2) = exp 1
2(x1u1+x2u2)e1e2
(27) (compare with equation (5)). In (24) the two frequencies u3 andu4are chosen to be zero, this corresponds to the fact that the involved rotations inR4 are isocline (see [22]).
Applications to color image processing are described in [1].
See also [24] for the construction of generalized color Fourier descriptors.
The main drawback of this transform is that it is a global transform with a fixed parametrizing bivector that does not take into account the geometric data.
III. THEGEOMETRICALFRAMEWORK
TO tackle this problem, we propose in this section to introduce a geometrical framework which allows for a pointwise version of the above transform. We refer the reader to [18] and [21] for the mathematical definitions and results used in particular in Sec. III. B.
A. From functions to surfaces Let
f : (x1, x2)∈Ω7−→f(x1, x2) (28) be a grey-level image defined on a domainΩofR2. Such an image can be considered as a surface Σembedded inR3 by the parametrization
ϕ: (x1, x2)7−→(x1, x2, f(x1, x2)) (29) that is as a 2-dimensional Riemannian surface with a global chart(Ω, ϕ). The Riemannian metric on Σis the one induced by the Euclidean metric ofR3.
Looking at formula (24) it seems natural to try to replace the parametrizing bivector mentioned above by a unit bivector field coding the tangent planes toΣ. This is the simplest way to involve the geometry ofΣ. But there are now two problems to deal with.
1) This bivector field (of the Clifford algebra R3,0) is clearly no longer constant.
2) To define the ⊥ action in this context, one has to introduce a varying space of representation.
In the same way, if
f : (x1, x2)∈Ω7−→(f1(x1, x2), f2(x1, x2), f3(x1, x2)) (30)
is a color image, it can be considered as a surfaceΣembedded inR5 by the parametrization
ϕ: (x1, x2)7→(x1, x2,(f1(x1, x2), f2(x1, x2), f3(x1, x2)) (31) and in this case one has to deal with varying bivectors of the Clifford algebra R5,0.
Solving the problems listed above requires the introduction of some advanced mathematical concepts (such as spinor bundles). The reader not familiar with these concepts can skip this part of the paper since we explain later how to compute practically the new defined spinor Fourier transform.
Let us just mention that because of the desire to use complex fast Fourier transforms we are led to consider surfaces as embedded in R4 for grey-level images and in R6 for color images.
B. Images as sections of associated bundles
LetΩbe an open set ofR2 (the domain of the image). Let PSO(En(Ω)) be the principal SO(n,R)-bundle of oriented frames of the trivial bundle En(Ω) = Ω× Rn. A spin structure on En(Ω) is a principal Spin(n)-bundle, denoted PSpin(En(Ω)), together with a 2-sheeted covering
PSpin(En(Ω))−→PSO(En(Ω)) (32) that is compatible with SO(n,R) and Spin(n) actions. We consider the following action
Spin(n)×PSpin(En(Ω))×Rn−→PSpin(En(Ω))×Rn (33) given by
(τ, p, z)7−→(pτ−1, ρn(τ)z) (34) where
ρn:Spin(n)−→SO(n,R) (35) is the standard representation of the group Spin(n) (see Appendix A). The associated bundlePSpin(En(Ω))×ρnRnis the quotient of the productPSpin(En(Ω))×Rn by the action (33). It is a vector bundle over Ωwith fiberRn.
Let now
f : (x1, x2)∈Ω7−→f(x1, x2) (36) be a grey-level image. Such an image is considered as a section of the associated bundlePSpin(E4(Ω))×ρ4R4, that is as a map σ: Ω−→PSpin(E4(Ω))×ρ4R4 (37) such that
π4◦σ=Id (38)
where
π4:PSpin(E4(Ω))×ρ4R4−→Ω (39) is the vector bundle projection. This section is given by
σ(x1, x2) =f(x1, x2)e3 (40) In the same way, to a color image
f : (x1, x2)∈Ω7−→(f1(x1, x2), f2(x1, x2), f3(x1, x2)) (41)
corresponds the section
σ(x1, x2) =f1(x1, x2)e3+f2(x1, x2)e4+f3(x1, x2)e5 (42) of the associated vector bundle PSpin(E6(Ω))×ρ6R6. At a given point (x1, x2)of Ω the value of the sectionσ belongs to a representation space on which the spin group consider as the fiber ofPSpin(En(Ω))at (x1, x2)acts. This allows to define a new spinor Fourier transform.
Remark. This geometric description applies for n- dimensional images,n≥3. As we will se later, the necessity of splitting the new defined transform onto orthogonal planes to apply complex fast Fourier transforms, impose here to treat the casesn= 4 andn= 6.
C. Spin characters and tangent planes
Let us first describe the spin characters involved (through the sections of the bundle PSpin(En(Ω)) in the definition of the spinor Fourier transform (see [1] for an alternative approach of the proofs).
Theorem 1: The morphisms from the additive groupR2 to the spin groupSpin(4)(theSpin(4)characters) are given by
(x1, x2)7−→exp1 2
(B1 B2)A x1
x2
(43) whereAis a2×2 real matrix (the frequency matrix) and
Bi=eifi (44)
for i= 1,2, with(e1, e2, f1, f2)an orthonormal basis ofR4. Proof:it relies on the following arguments.
1) R2 is simply connected so that every Lie group mor- phism from R2 to Spin(4) is given by the exponenti- ation of a Lie algebra homomorphism from R2 to the Lie algebra spin(4)of Spin(4).
2) The groupSpin(4) is isomorphic to the group product Spin(3)×Spin(3).
3) The abelian Lie subalgebras of the Lie algebraspin(3) of the group Spin(3) are of dimension 1.
4) The orthogonalization algorithm of Hestenes on bivec- tors (see [20]) allows to express the morphisms fromR2 toSpin(3)×Spin(3)as in formula (44).
ASpin(4) character is then parametrized by 4 real numbers (the frequencies) and an orthonormal basis of R4.
Theorem 2: The morphisms from the additive groupR2 to the spin groupSpin(6)(theSpin(6)characters) are given by
(x1, x2)7−→exp1 2
(B1 B2 B3)A x1
x2
(45) whereAis a3×2 real matrix (the frequency matrix) and
Bi=eifi (46)
fori= 1,2,3, with(e1, e2, e3, f1, f2, f3)an orthonormal basis of R6.
Proof:as in the proof of Theorem 1, we have to describe the Lie algebra homomorphisms fromR2 to the Lie algebra spin(6)of the groupSpin(6). This is done using the following results:
1) Spin(6) is isomorphic toSU(4): consider
Spin(6)⊂R(0)6,0'R5,0'C(4) (47) so that Spin(6) appears as a subgroup of GL(4,C) which implies by standard arguments (compactness and connectedness) the isomorphism.
2) As a consequence, the Lie algebra spin(6) of Spin(6) is of typeE(3) so that it contains maximal abelian Lie subalgebras of dimension 3.
3) The bivectors Bi form bases of such Lie sub-algebras (Cartan subalgebras).
ASpin(6)character is then parametrized by 6 real numbers (the frequencies) and an orthonormal basis of R6.
In the rest of this paper, the spin characters (43) and (45) are denoted χA,B (A is the involved frequency matrix and B stands for (B1, B2) or (B1, B2, B3) depending on the context). Considering images as sections of the bundle PSpin(En(Ω))×ρnRn (n= 4or n= 6) allows to deal with actions of sections of the bundle PSpin(En(Ω)). This means thatB is no longer fixed and may depend of the current point.
Let now
ϕ: (x1, x2)7−→(x1, x2, f(x1, x2)) (48) be the graph of a grey-level image. At each pointpofΩ, the space R3 splits into
R3=Tϕ(p)Σ⊕Nϕ(p)Σ (49) where TΣ, resp. NΣ, denotes the tangent, resp. the normal, bundle of the surfaceΣ. LetF4 be the bundle
F4=TΣ⊕NΣ⊕Re4 (50) andCl(F4)be the corresponding Clifford bundle. The degree one sections ofCl(F4)can be identified with functions from Ω toR4 and as mentioned before, we consider the image as a section σ(x1, x2) = f(x1, x2)e3 of Cl(F4). The tangent bundle TΣ is encoded by the degree 2 section (the bivector field)
B1= ϕx1∧ϕx2
kϕx1∧ϕx2k (51) of Cl(F4). Here ϕxi denotes the partial derivative of ϕwith respect to xi. More precisely,
B1=γ1e1e2+γ2e1e3+γ3e2e3 (52) where
γ1= 1
p1 +fx2
1+fx2
2
γ2= fx2
p1 +fx2
1+fx2
2
(53) and
γ3= −fx1
p1 +fx21+fx22 (54) In this case B2 is given by I4B1 (I4 is the pseudo scalar of the Clifford algebra R4,0), that is
B2=−γ3e1e4+γ2e2e4−γ1e3e4 (55)
In the same way, if
ϕ: (x1, x2)7→(x1, x2, f1(x1, x2), f2(x1, x2), f3(x1, x2)) (56) is the graph of a color image, we introduce the decomposition R5=Tϕ(p)Σ⊕Nϕ(p)Σ (57) the bundle
F6=TΣ⊕NΣ⊕Re6 (58) the Clifford bundleCl(F6)and identify a color image with a degree one sectionσ(x1, x2) =f1(x1, x2)e3+f2(x1, x2)e4+ f3(x1, x2)e5 ofCl(F6). In this case the bivectorB1 reads
B1=γ1e1e2+γ2e1e3+γ3e1e4+γ4e1e5+γ5e2e3 +γ6e2e4+γ7e2e5+γ8e3e4+γ9e3e5+γ10e4e5 (59) where
γ1= 1/δ γ2= (f1)x2/δ γ3= (f2)x2/δ γ4= (f3)x2/δ γ5= (−f1)x1/δ γ6= (−f2)x1/δ γ7= (−f3)x1/δ
γ8= [(f1)x1(f2)x2−(f1)x2(f2)x1]/δ γ9= [(f1)x1(f3)x2−(f1)x2(f3)x1]/δ
γ10= [(f2)x1(f3)x2−(f2)x2(f3)x1]/δ (60) with δ =kϕx1 ∧ϕx2k. The bivectors B2 andB3 are deter- mined using Gram-Schmidt algorithm (see remark 3 below).
D. The spinor Fourier transform
We follow the same procedure as in Sec. II.B. Let σ be a section of the bundle PSpin(En(Ω))×ρnRn (n = 4 or n = 6) and χA,B be a section of PSpin(En(Ω)) where B = B(x1, x2) is a fixed field of bivectors that satisfies for each (x1, x2) of Ω the conditions (44) or (46), i.e Bi(x1, x2) =ei(x1, x2)fi(x1, x2), the vectorsei(x1, x2)and fi(x1, x2)being orthonormal.
Before to give the precise definition, let us make an impor- tant remark. As we want to define an invertible transform, this one has to preserve the fibered structure we have introduced.
This means that we need to distinguish(x1, x2)as the variable of the image (or of the corresponding section) and(x1, x2)as the position of the fiber. To do this we introduce
eσ(x1, x2, y1, y2) = X
i
δi(y1, y2)ei(x1, x2) +X
i
τi(y1, y2)fi(x1, x2) (61) where
δi(y1, y2) =σ(y1, y2)·ei(y1, y2) (62) and
τi(y1, y2) =σ(y1, y2)·fi(y1, y2) (63) Definition 1 (Spinor Fourier transform): with the previous notations, the spinor Fourier transform of σ = σ(y1, y2) is given by
FBσ(A) =
